It sounds like the problem you're having is that some destination website doesn't want to receive connections from some of the Tor exit relays. For more on that issue, you might find this blog post educational:
https://blog.torproject.org/blog/call-arms-helping-internet-services-ac…

But it is a different issue than what you call the 'base relay', and what we call the guard relay. The exit relay is the third hop in the circuit, which destination websites see. The guard relay is the first hop in the circuit, and destination websites don't see it. So I think they are unrelated.